Critical $Sp(N)$ Models in $6-\epsilon$ Dimensions and Higher Spin dS/CFT

TL;DR

This paper explores non-unitary $Sp(N)$ scalar field theories in 6-$\e$ dimensions, revealing fixed points, operator dimensions, and potential dualities with higher spin de Sitter theories, with implications for non-unitary conformal field theories.

Contribution

It introduces a new class of non-unitary $Sp(N)$ models in 6-$\e$ dimensions, analyzes their fixed points and operator dimensions, and proposes dualities with higher spin de Sitter theories.

Findings

Existence of IR stable fixed points at imaginary couplings for even $N$.

Development of $\e$ expansions for operator dimensions and free energy.

Identification of enhanced symmetry $OSp(1|2)$ at $N=2$ and connection to Potts model.

Abstract

Theories of anti-commuting scalar fields are non-unitary, but they are of interest both in statistical mechanics and in studies of the higher spin de Sitter/Conformal Field Theory correspondence. We consider an $Sp(N)$ invariant theory of $N$ anti-commuting scalars and one commuting scalar, which has cubic interactions and is renormalizable in 6 dimensions. For any even $N$ we find an IR stable fixed point in $6-\epsilon$ dimensions at imaginary values of coupling constants. Using calculations up to three loop order, we develop $\epsilon$ expansions for several operator dimensions and for the sphere free energy $F$. The conjectured $F$-theorem is obeyed in spite of the non-unitarity of the theory. The $1/N$ expansion in the $Sp(N)$ theory is related to that in the corresponding $O(N)$ symmetric theory by the change of sign of $N$. Our results point to the existence of interacting non-unitary 5-dimensional CFTs with $Sp(N)$ symmetry, where operator dimensions are real. We conjecture that these CFTs are dual to the minimal higher spin theory in 6-dimensional de Sitter space with Neumann future boundary conditions on the scalar field. For $N=2$ we show that the IR fixed point possesses an enhanced global symmetry given by the supergroup $OSp(1|2)$. This suggests the existence of $OSp(1|2)$ symmetric CFTs in dimensions smaller than 6. We show that the $6-\epsilon$ expansions of the scaling dimensions and sphere free energy in our $OSp(1|2)$ model are the same as in the $q \rightarrow 0$ limit of the $q$-state Potts model.